by the (bivariate) Mean Value Theorem, where \(0\le p_i\le P_i\)  and \(0\le p_r\le P_r\) , see [1]}.
Defining \(\bar{C}:=\max _{n\ge 1}\max _{(p_i,p_r)\in \mathcal {D}^{\lbrace i,r\rbrace }_\delta }\left|\frac{\partial ^2\Psi _n}{\partial P_i\partial P_r}(p_i,p_r)\right|\) , we finally conclude \(|\Psi _n(P_i,P_r)|\le \bar{C}P_iP_r\)  for all \(n\ge 1\)  and \((P_i,P_r)\in \mathcal {D}^{\lbrace i,r\rbrace }_\delta \) , establishing the uniformity claim.
By our asymptotics for \(\lambda _1\) , we similarly obtain
\(\ln \lambda _1(P_i,P_r)-\ln \lambda _1(P_i,0)-\ln \lambda _1(0,P_r)+\ln \lambda _1(0,0)=P_iP_r\frac{\partial ^2\ln \lambda _1}{\partial P_i\partial P_r}(p_i,p_r)=\mathcal {O}(P_iP_r(P_i+P_r)^2),\) 
